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Menz, Georg; Otto, Felix

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. (English) Zbl 1282.60096

Ann. Probab. 41, No. 3B, 2182-2224 (2013).
The paper addresses a noninteracting unbounded spin system with the mean spin conservation constraint. For the canonical ensemble, with the choice of super-quadratic single-site potentials, an old Varadhan’s problem [S. R. S. Varadhan, Pitman Res. Notes Math. Ser. 283, 75–128 (1993; Zbl 0793.60105)] is addressed of the validity fo the spectral gap inequality and next its strenghthening by means of the logarithmic Sobolev inequality. Since standard criterions to this end (e.g. the tensorization principle, Bakry-Émery and Holley-Stroock criterions) fail for the canonical ensemble, new techniques are developed. They stem from the previous (co-authored by F. Otto) work on the two-scale approach [N. Grunewald et al., Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 2, 302–351 (2009; Zbl 1179.60068)]. Using the asymmetric Brascamp-Lieb type inequality for covariances, the task of deriving a unform logarithmic Sobolev inequality is reduced to the convexification of the coarse grained Hamiltonian by iterated renormalization. A local Cramer theorem is employed, the idea bororowed from [N. Grunewald et al., loc. cit.]
Reviewer: Piotr Garbaczewski (Opole)

Cited in 2 Reviews
Cited in 23 Documents

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Keywords:

canonical ensemble; spectral gap inequality; logarithmic Sobolev inequality; spin system; Kawasaki dynamics; single site potential; adapted two-scale approach; iterated renormalization

Citations:

Zbl 0793.60105; Zbl 1179.60068
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\textit{G. Menz} and \textit{F. Otto}, Ann. Probab. 41, No. 3B, 2182--2224 (2013; Zbl 1282.60096)
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References:

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